introduction to functional programming in...

Introduction to Functional Programming in Haskell

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Outline

Why learn functional programming?

The essence of functional programmingWhat is a function?Equational reasoningFirst-order vs. higher-order functionsLazy evaluation

How to functional programFunctional programming workflowData typesType classesType-directed programmingHaskell styleRefactoring (bonus section)

Type inference

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Outline


The essence of functional programming

How to functional program

Type inference

Why learn functional programming? 3 / 55

Why learn (pure) functional programming?

1. This course: strong correspondence of core concepts to PL theory• abstract syntax can be represented by algebraic data types• denotational semantics can be represented by functions

2. It will make you a better (imperative) programmer• forces you to think recursively and compositionally• forces you to minimize use of state

. . . essential skills for solving big problems

3. It is the future!• more scalable and parallelizable• functional features have been added to most mainstream languages• many cool new libraries built around functional paradigm


Quotes from pragmatists

Programming is not about data. It’s about transforming data . . .That’s why I think functional programming is a natural successor toobject-oriented programming. In OOP, we’re constantly concerned aboutthe state of our data. In functional programming, our focus is on thetransformation of data. And transformation is where the value is added.

—Dave Thomas, The Pragmatic Programmer



Without understanding functional programming, you can’t invent MapReduce,the algorithm that makes Google so massively scalable.

—Joel Spolsky, Joel on Software



So that’s the big deal about functional languages; and it is one big frickingdeal. There is a freight train barreling down the tracks towards us, withmulti-core emblazoned on it; and you’d better be ready by the time it gets here.

—Robert C. Martin, a.k.a. “Uncle Bob”


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Type inference

The essence of functional programming 8 / 55

What is a (pure) function?

A function is pure if:• it always returns the same output for the same inputs• it doesn’t do anything else — no “side effects”

In Haskell: whenever we say “function” we mean a pure function!


What are and aren’t functions?

Always functions:• mathematical functions f (x) = x2 + 2x + 3• encryption and compression algorithms

Usually not functions:• C, Python, JavaScript, . . . “functions” (procedures)• Java, C#, Ruby, . . . methods

Haskell only allows you to write (pure) functions!


Why procedures/methods aren’t functions

• output depends on environment• may perform arbitrary side effects


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Type inference


Getting into the Haskell mindset

Haskellsum :: [Int] -> Intsum [] = 0sum (x:xs) = x + sum xs

Javaint sum(List<Int> xs) {int s = 0;for (int x : xs) {s = s + x;

}return s;

}In Haskell, “=” means is not change to!


Getting into the Haskell mindset

Quicksort in Haskellqsort :: Ord a => [a] -> [a]qsort [] = []qsort (x:xs) = qsort (filter (<= x) xs)

++ [x] ++ qsort (filter (> x) xs)

Quicksort in Cvoid qsort(int low, int high) {int i = low, j = high;int pivot = numbers[low + (high-low)/2];

while (i <= j) {while (numbers[i] < pivot) {i++;

}while (numbers[j] > pivot) {j--;

}if (i <= j) {swap(i, j);i++;j--;

}}if (low < j)qsort(low, j);

if (i < high)qsort(i, high);

}

void swap(int i, int j) {int temp = numbers[i];numbers[i] = numbers[j];numbers[j] = temp;

}


Referential transparency

An expression can be replaced by its valuewithout changing the overall program behavior

length [1,2,3] + 4⇒ 3 + 4

what if length was a Java method?

Corollary: an expression can be replaced by any expressionwith the same value without changing program behavior

Supports equational reasoning


a.k.a. referent

Equational reasoning

Computation is just substitution!

sum :: [Int] -> Intsum [] = 0sum (x:xs) = x + sum xs

equations

sum [2,3,4]

⇒ sum (2:(3:(4:[])))

⇒ 2 + sum (3:(4:[]))

⇒ 2 + 3 + sum (4:[])

⇒ 2 + 3 + 4 + sum []

⇒ 2 + 3 + 4 + 0

⇒ 9


Describing computations

Function definition: a list of equations that relate inputs to output• matched top-to-bottom• applied left-to-right

Example: reversing a listimperative view: how do I rearrange the elements in the list? 7functional view: how is a list related to its reversal? 3

reverse :: [a] -> [a]reverse [] = []reverse (x:xs) = reverse xs ++ [x]


Exercise

1. Evaluate: double (succ (double 3))

A : 12 B : 14 C : 16

2. Prove, up to associativity of (+), using equational reasoning:

double (succ x) = succ (succ (double x))


succ :: Int -> Intsucc x = x + 1

double :: Int -> Intdouble x = x + x

Exercise

1. Evaluate: foo (foo 0 0) 0

A : 0 B : 2 C : 3

2. Evaluate: foo 0 (foo 0 0)

A : 0 B : 2 C : 3


foo :: Int -> Int -> Intfoo 0 x = x + 1foo x 0 = x + 2foo x y = x + y

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Type inference


First-order functions

Examples• cos :: Float -> Float• even :: Int -> Bool• length :: [a] -> Int


Higher-order functions

Examples• map :: (a -> b) -> [a] -> [b]• filter :: (a -> Bool) -> [a] -> [a]• (.) :: (b -> c) -> (a -> b) -> a -> c


Higher-order functions as control structures

map: loop for doing something to each element in a list

map :: (a -> b) -> [a] -> [b]map f [] = []map f (x:xs) = f x : map f xs

map f [2,3,4,5] = [f 2, f 3, f 4, f 5]

map even [2,3,4,5]= [even 2, even 3, even 4, even 5]= [True,False,True,False]

fold: loop for aggregating elements in a list

foldr :: (a->b->b) -> b -> [a] -> bfoldr f y [] = yfoldr f y (x:xs) = f x (foldr f y xs)

foldr f y [2,3,4] = f 2 (f 3 (f 4 y))

foldr (+) 0 [2,3,4]= (+) 2 ((+) 3 ((+) 4 0))= 2 + (3 + (4 + 0))= 9


Function composition

Can create new functions by composing existing functions• apply the second function, then apply the first

Function composition(.) :: (b -> c) -> (a -> b) -> a -> cf . g = \x -> f (g x)

(f . g) x = f (g x)

Types of existing functionsnot :: Bool -> Boolsucc :: Int -> Inteven :: Int -> Boolhead :: [a] -> atail :: [a] -> [a]

Definitions of new functionsplus2 = succ . succodd = not . evensecond = head . taildrop2 = tail . tail


Currying / partial application

In Haskell, functions that take multiple arguments areimplicitly higher order

plus :: Int -> Int -> Int

increment :: Int -> Intincrement = plus 1

Curried plus 2 3plus :: Int -> Int -> Int

Uncurried plus (2,3)plus :: (Int,Int) -> Int

a pair of intsThe essence of functional programming 25 / 55

Haskell Curry

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Type inference


Lazy evaluation

In Haskell, expressions are reduced:• only when needed• at most once

Supports:• infinite data structures• separation of concerns

nats :: [Int]nats = 1 : map (+1) nats

fact :: Int -> Intfact n = product (take n nats)

min3 :: [Int] -> [Int]min3 = take 3 . sort

What is the running time of this function?


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Type inference

How to functional program 28 / 55

Functional programming workflow

“obsessive compulsive refactoring disorder”


FP workflow (detailed)


Norman Ramsey, On Teaching “How to Design Programs”, ICFP’14

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Type inference


Algebraic data types

Data type definition• introduces new type of value• enumerates ways to constructvalues of this type

Some example data typesdata Bool = True | False

data Nat = Zero | Succ Nat

data Tree = Node Int Tree Tree| Leaf Int

Definitions consists of . . .• a type name• a list of data constructorswith argument types

Definition is inductive• the arguments may recursivelyinclude the type being defined

• the constructors are the only wayto build values of this type


Anatomy of a data type definition

type name

data Expr = Lit Int| Plus Expr Expr cases

data constructor types of arguments

Example: 2+ 3+ 4 Plus (Lit 2) (Plus (Lit 3) (Lit 4))


FP data types vs. OO classes

Haskelldata Tree = Node Int Tree Tree

| Leaf

• separation of type- and value-level• set of cases closed• set of operations open

Javaabstract class Tree { ... }class Node extends Tree {int label;Tree left, right;...

}class Leaf extends Tree { ... }

• merger of type- and value-level• set of cases open• set of operations closed

Extensibility of cases vs. operations = the “expression problem”


Type parameters

(Like generics in Java)type parameter

data List a = Nil| Cons a (List a)

reference totype parameter

recursivereference to type

Specialized liststype IntList = List Inttype CharList = List Chartype RaggedMatrix a = List (List a)


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Type inference


What is a type class?

1. an interface that is supported by many different types2. a set of types that have a common behavior

class Eq a where(==) :: a -> a -> Bool

types whose values can becompared for equality

class Show a whereshow :: a -> String

types whose values can beshown as strings

class Num a where(+) :: a -> a -> a(*) :: a -> a -> anegate :: a -> a...

types whose values can bemanipulated like numbers


Type constraints

List elements can be of any typelength :: [a] -> Intlength [] = 0length (_:xs) = 1 + length xs

List elements must support equality!elem :: Eq a => a -> [a] -> Boolelem _ [] = Falseelem y (x:xs) = x == y || elem y xs

use method⇒ add type class constraint


class Eq a where(==) :: a -> a -> Bool

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Type inference


Tools for defining functions

Recursion and other functionssum :: [Int] -> Intsum xs = if null xs then 0

else head xs + sum (tail xs)

(1) case analysis

Pattern matchingsum :: [Int] -> Intsum [] = 0sum (x:xs) = x + sum xs

(2) decompositionHigher-order functionssum :: [Int] -> Intsum = foldr (+) 0

no recursion or variables needed!How to functional program 40 / 55

What is type-directed programming?


Use the type of a functionto help write its body

Type-directed programming

Basic goal: transform values of argument types into result type

If argument type is . . .

• atomic type (e.g. Int, Char)• apply functions to it

• algebraic data type• use pattern matching

• case analysis• decompose into parts

• function type• apply it to something

If result type is . . .• atomic type

• output of another function

• algebraic data type• build with data constructor

• function type• function composition orpartial application

• build with lambda abstraction


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Type inference


Good Haskell style

Why it matters:• layout is significant!• eliminate misconceptions• we care about elegance

Easy stuff:• use spaces! (tabs cause layout errors)• align patterns and guards

See style guides on course web page


Formatting function applications

Function application:• is just a space• associates to the left• binds most strongly

f(x) f x

(f x) y f x y

(f x) + (g y) f x + g y

Use parentheses only to override this behavior:• f (g x)• f (x + y)


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Type inference


Refactoring in the FP workflow

“obsessive compulsive refactoring disorder”

Motivations:• separate concerns• promote reuse• promote understandability• gain insights


Refactoring relations

Semantics-preserving laws prove with equational reasoning and/or induction

• Eta reduction:\x -> f x ≡ f

• Map–map fusion:map f . map g ≡ map (f . g)

• Fold–map fusion:foldr f b . map g ≡ foldr (f . g) b

“Algebra of computer programs”

John Backus, Can Programming be Liberated from the von Neumann Style?, ACM Turing Award Lecture, 1978


Strategy: systematic generalization

commas :: [String] -> [String]commas [] = []commas [x] = [x]commas (x:xs) = x : ", " : commas xs

Introduce parameters for constantsseps :: String -> [String] -> [String]seps _ [] = []seps _ [x] = [x]seps s (x:xs) = x : s : seps s xs

Broaden the typesintersperse :: a -> [a] -> [a]intersperse _ [] = []intersperse _ [x] = [x]intersperse s (x:xs) = x : s : intersperse s xs


commas :: [String] -> [String]commas = intersperse ", "

Strategy: abstract repeated templates

abstract (v): extract and make reusable (as a function)

showResult :: Maybe Float -> StringshowResult Nothing = "ERROR"showResult (Just v) = show v

moveCommand :: Maybe Dir -> CommandmoveCommand Nothing = StaymoveCommand (Just d) = Move d

safeAdd :: Int -> Maybe Int -> IntsafeAdd x Nothing = xsafeAdd x (Just y) = x + y

Repeated structure:• pattern match• default value if Nothing• apply function to contents if Just


Strategy: abstract repeated templates

Describe repeated structure in functionmaybe :: b -> (a -> b) -> Maybe a -> bmaybe b _ Nothing = bmaybe _ f (Just a) = f a

Reuse in implementationsshowResult = maybe "ERROR" showmoveCommand = maybe Stay MovesafeAdd x = maybe x (x+)


Refactoring data types

data Expr = Var Name| Add Expr Expr| Sub Expr Expr| Mul Expr Expr

vars :: Expr -> [Name]vars (Var x) = [x]vars (Add l r) = vars l ++ vars rvars (Sub l r) = vars l ++ vars rvars (Mul l r) = vars l ++ vars r

eval :: Env -> Expr -> Inteval m (Var x) = get x meval m (Add l r) = eval m l + eval m reval m (Sub l r) = eval m l - eval m reval m (Mul l r) = eval m l * eval m r


Refactoring data types

Factor out shared structuredata Expr = Var Name

| BinOp Op Expr Expr

data Op = Add | Sub | Mul

vars :: Expr -> [Name]vars (Var x) = [x]vars (BinOp _ l r) = vars l ++ vars r

eval :: Env -> Expr -> Inteval m (Var x) = get x meval m (BinOp o l r) = op o (eval m l) (eval m r)whereop Add = (+)op Sub = (-)op Mul = (*)


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Type inference

Type inference 54 / 55

Type inference

How to perform type inferenceIf a literal, data constructor, or named function: write down the type – you’re done!

Otherwise:1. expand any syntactic sugar2. pick an application e1 e23. recursively infer their types e1 : T1 and e2 : T24. T1 should be a function type T1 = Targ → Tres5. unify Targ =? T2, yielding type variable assignment σ6. return e1 e2 : σTres (Tres with type variables substituted)If any of these steps fails, it is a type error!

Type inference 55 / 55

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